Given each set of vertices, determine whether PQRS is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning, P(-2, -3). Q(2, - 6). R(6. - 3). S(2, 1)

Step-by-step explanation:

Given the coordinates P(-2, -3). Q(2, - 6). R(6. - 3). S(2, 1), to determine the type of shape the quadrilateral is, we need to find the measure of the sides. To get the measure of each sides, we will take the distance between the adjacent coordinates using the formula to formula for calculating the distance between two points as shown;

D = √(x₂-x₁)²-(y₂-y₁)²

For the side PQ with the coordinate P(-2, -3). Q(2, - 6)

PQ = √(2-(-2))²-(-6-(-3))²

PQ = √(2+2)²-(-6+3)²

PQ = √4²-(-3)²

PQ = √16-9

PQ = √7

For the side QR with the coordinate Q(2, - 6) and R(6, -3)

QR = √(6-2))²-(-3-(-6))²

QR = √(4)²-(3)²

QR = √16-9

QR = √7

For the side RS with the coordinate R(6. - 3) and S(2, 1)

RS = √(2-6)²-(1-(-3))²

RS = √(-4)²-(1+3)²

RS = √16-(4)²

RS = √16-16

RS = 0

For the side PS with the coordinate P(-2, -3) and S(2, 1)

PS = √(2-(-2))²-(1-(-3))²

PS = √(4)²-(1+3)²

PS = √16-(4)²

PS = √16-16

PS = 0

For the quadrilateral to be a rectangle, then two of its sides must be equal and parallel to each other. A rectangle is a plane shape that has two of its adjacent sides equal and parallel to each other. Since two of he sides are equal i.e RS = PS and PQ = QR then the quadrilateral PQRS is a rectangle. Both rhombus and square has all of its sides equal thereby making them wrong.